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page 12(Stein and Shakarchi, Complex Analysis

page 13(Stein and Shakarchi, Complex Analysis

My question has to do with the proof for proposition $2.3$. I am wondering how he arrived at the first equation on Page $13$. Can someone explain the complete proof?

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  • $\begingroup$ Two by two matrices of the form $\begin{pmatrix} a & b \\ -b & a \end{pmatrix}$ with real entries are isomorphic to the complex numbers. $\endgroup$ Aug 29 '20 at 2:24
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I am not sure whether my understanding is the true...

I think in the first question on Page 13, the left side $J_F(x_0,y_0)(H)$ is a matrix multiplication (it may be better if $H$ is written into column vector)...And the middle side is in 'complex number form'...

So I think the first '$=$' should be a kind of isomorphism...

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